Lesson Video: Multiplying Monomials Mathematics

In this video, we will learn how to multiply monomials involving single and multiple variables.

13:26

Video Transcript

In this video, we’ll learn how to multiply monomials involving single and multiple variables. We’ll begin, of course, by recalling what we actually mean by the term monomial.

The word mono- is from the ancient Greeks. “Monos” meant alone, only, solo, or single. And so, we say a monomial is a polynomial with just one term, in other words, a single term that’s made up of numbers and/or variables. They might have multiple variables, but the exponents of any part of the term will only ever be positive integers. For instance, three 𝑥 to the fourth power 𝑦 is a monomial. But two 𝑥 to the power of negative five is not, since its exponent is negative. Similarly, a radical or a surd is not considered to be a monomial. And this is because we can represent radicals as fractional exponents. And of course, a fraction is not an integer. So, we’re going to look at multiplying two or more monomials. And we’re going to begin simply by looking at a problem involving no algebraic terms.

Find the value of three times three cubed times three squared.

We’re being asked to find the value of this product. And so, what we could do is begin by working out the value of three cubed and three squared. Three squared, of course, is the same as three times three, which is equal to nine. And then, three cubed is three times three times three, which is equal to 27. And so, to find the value of three times three cubed times three squared, we’re going to work out three times 27 times nine. Now, of course, multiplication is commutative. It can be performed in any order. So, let’s begin by working out three times nine. Three multiplied by nine is 27. So, three times 27 times nine is the same as 27 times 27. And of course, we can use any method to calculate this.

Let’s recall how we might use the column method. Seven times seven is 49. So, we put a nine here, and we carry the four. Two times seven is 14, and then when we add this four, we get 18. And so, we found that 27 times seven is 189. We’re now going to multiply the digits two and seven by this two. Now, of course, this two is in the tens column, so it’s equivalent to multiplying by 20. And so, we can add a zero here to take this into account. Seven times two is 14. So, we put a four here and carry the one. Then, two times two is four, and when we add this one, we get five. So, 27 times 20 is 540. We’re now going to add these two numbers. Nine plus zero is nine; eight plus four is 12. And so, we carry the one. Then, one plus five plus that carried one is seven. And so, 27 times 27 is 729. And of course, we can therefore say that that’s the value of our product.

But was there another way we could’ve done this? Well, yes, we have a rule for working with exponential terms. As long as the base is the same, to multiply these sorts of terms, we simply add their exponents. The base in this general rule is 𝑥. And so, 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. And we can extend this rule into multiplying three terms. If we think of three as three to the first power, we can say that three to the first power times three cubed times three squared is the same as three to the power of one plus three plus two, which is three to the sixth power. Now, of course, we’ve really just simplified our product. We do need to still evaluate it to get 729. Either way, we find the value of three times three cubed times three squared to be equal to 729.

And so, we’ve looked at how we can simplify and evaluate products of monomials made up of purely numerical terms. So, what do we do when we work with products of numbers and variables?

Remember this dot is used interchangeably with the multiplication symbol. And so, to simplify the expression given, we’re going to recall some of the properties of multiplication. Firstly, the associative property of multiplication. This says that when three or more numbers are multiplied, the product is the same, no matter how those numbers are grouped. And so, we don’t really need the parentheses here. It will be the same as just doing six times 𝑥 times eight. Then, the commutative property says that the product will be the same even if we switch the numbers around. For instance, three times four is the same as four times three. And so, we’re going to switch the 𝑥 and the eight around and rewrite our problem as six times eight times 𝑥. And then, of course, since the grouping doesn’t really matter, we can evaluate six times eight. Six times eight is 48, so this becomes 48 times 𝑥. And of course, we know that we can simply write that as 48𝑥. So, when we simplify the expression six times eight times 𝑥, we get 48𝑥.

And so, we’ve seen now how to multiply fairly simple monomials using some of the multiplication properties, along with the laws for working with exponents. We’re now going to combine these and look at some more complicated problems.

Simplify seven 𝑥 to the fourth power times eight 𝑥 to the seventh power.

There are two things we know about multiplication. It is associative and commutative. The associative property tells us that the product will be the same no matter how the numbers are grouped. And the commutative property tells us that the order in which we perform the calculation really doesn’t matter. And so, we’re going to begin by sort of unsimplifying each term in our expression. When we do, we get seven times 𝑥 to the fourth power times eight times 𝑥 to the seventh power. We’re now going to switch around 𝑥 to the fourth power and eight. And so, we see that our expression now becomes seven times eight times 𝑥 to the fourth power times 𝑥 to the seventh power. And in fact, we can work out the numerical part. We know that seven times eight is 56, which means that this becomes 56 times 𝑥 to the fourth power times 𝑥 to the seventh power.

But what do we do with these algebraic parts? Well, we have a rule for multiplying exponential terms. As long as the base is the same, we add the exponents. So, 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. This in turn means that 𝑥 to the fourth power times 𝑥 to the seventh power is 𝑥 to the power of four plus seven. And since four plus seven is 11, this becomes 𝑥 to the 11th power. Now, in fact, we know that we don’t really want to include that multiplication symbol. And so, we simplify this further to get 56𝑥 to the 11th power. And so, when we simplify seven 𝑥 to the fourth power times eight 𝑥 to the seventh power, we get 56𝑥 to the 11th power.

Now, actually, all this sort of unsimplifying and then switching terms around is quite long winded, so we can actually generalize. We say that to multiply two or more monomials, we can first multiply the coefficients, those are the numbers in front of the algebraic parts, and then separately multiply the variables, of course, using the laws of exponents when necessary. Let’s apply this to a context-based question to begin with.

Find an expression for the volume of the rectangular prism shown.

And then, we have a rectangular prism or a cuboid where we’ve been given the three dimensions. Let’s call this dimension here the length, which I’ve abbreviated to 𝑙. We’ll call this dimension here 𝑤 for width, and then we’ll say this dimension is the height ℎ. And then, we know, of course, that the volume of a rectangular prism is just the product of these three dimensions. It’s length times width times height, meaning that the volume of our rectangular prism in cubic units will be three 𝑥 times 10𝑥 times 15𝑥. And so, we’re actually finding the product of three monomials. And we recall that to do this, we first multiply the coefficients; those are the numerical parts. So, we’re going to begin by doing three times 10 times 15. And then, we separately multiply the variables. So here, we’re going to do 𝑥 times 𝑥 times 𝑥.

We do also know that multiplication is commutative, so it can be done in any order. So, we’ll begin by multiplying the three by the 15 to get 45. And then 45 multiplied by this 10 here is 450. And so, when we multiply the coefficients of our monomials, we get 450. And now, we multiply the algebraic parts. 𝑥 times 𝑥 times 𝑥 is 𝑥 cubed. And so, we can say that the volume of the rectangular prism shown in cubic units is 450𝑥 cubed.

Let’s now consider what happens when our monomials contain more than one variable.

Simplify three 𝑥 times four 𝑦.

Here, we’re looking to find the product of two monomials. The first is three 𝑥 and the second is four 𝑦. And so, we can recall that when we multiply monomials, we begin by multiplying the coefficients. That’s the numbers in front of the algebraic parts. And then, we can multiply the variables using our laws for exponents, if necessary. Now, the coefficients of our two monomials are three and four. So, we first work out three times four, which is equal to 12. Then, we see, of course, that the variable parts are 𝑥 and 𝑦. So, we do 𝑥 times 𝑦, which we just write as 𝑥𝑦. Three 𝑥 times four 𝑦 then will just be the product of these terms. It’s 12 times 𝑥𝑦 that we can write as 12𝑥𝑦. And so, we’ve simplified three 𝑥 times four 𝑦, and we got 12𝑥𝑦.

Let’s now combine everything we’ve done into looking at one slightly nastier example.

Simplify six 𝑥 squared 𝑦 squared times negative nine 𝑥 squared 𝑦 to the fifth power.

Here, we’re finding the product of two monomials. And so, we recall the process for multiplying two or more monomials. We begin by multiplying the coefficients of each term. And then we separately multiply the variables. Now, we will need to use the laws of exponents to do so. The specific law that we’re going to use is the fact that when the bases are the same, we’d multiply exponential terms by just adding the exponents. So, 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. Let’s identify the coefficients in our product. We have six here and negative nine here, and so we’re going to multiply six by negative nine. Six times nine is 54. And a positive multiplied by a negative is a negative, so six times negative nine is negative 54.

Next, we multiply the variables. We’re going to multiply 𝑥 squared by 𝑥 squared. And of course, to do so, we simply add the exponents. So, we get 𝑥 to the power of two plus two, which is, of course, the same as 𝑥 to the fourth power. There’s another variable we need to consider though, and that’s 𝑦. So, we do the same with this one. 𝑦 squared times 𝑦 to the fifth power is the same as 𝑦 to the power of two plus five. And of course, since two plus five is seven, 𝑦 squared times 𝑦 to the fifth power is 𝑦 to the seventh power. When we combine all of this, we get the result of multiplying six 𝑥 squared 𝑦 squared by negative nine 𝑥 squared 𝑦 to the fifth power. It’s negative 54𝑥 to the fourth power 𝑦 to the seventh power.

We’re now going to recap the key points from this lesson. In this video, we learned that a monomial is a polynomial with just one term. We know that that single term can be made up of numbers and/or variables, and it might even contain more than one variable, but that any exponents in that expression would always be positive integers. We would never include, say, a fractional exponent nor a negative one. We also saw that to multiply two or more monomials, we multiply the coefficients and then we separately multiply the variables using the laws of exponents where necessary. And the law for exponents that we used was 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. In other words, as long as the base is the same, we simply add the exponents.

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